Take an unital differential graded (dg) $k$-algebra $A$, we can regard it as $A_\infty$-algebra with $m_1$ as differential and $m_2$ as algebra multiplication, and $m_n=0$ or all $n\geq 0$. Take a dg $A$-module $M$, then we can form the $A_\infty$-algebra $B=End_A(M)$.
We say $B$ is formal if the homology $H^*(B)$ is quasi-isomorphic to $B$. Apparently, formality of $B$ implies derived (dg) equivalence, i.e. equivalence between the derived dg categories $D(A)$ and $D(B)$, and hence with $D(H^*(B))$. Is this true? Where can I find an exact reference which states anything like this? I have looked through Professor Keller's note "Introduction to $A_\infty$-algebra and modules", but doesn't seems to see anything like this.
Moreover, is the converse statement true? i.e. if $B$ is not formal, then there is no derived equivalence between $D(A)$ and $D(H^*(B))$.?